In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z (x, ω)) where x is the spatial variable and Z (x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of a. Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen-Loève expansion for computing realizations of a (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based

Machine learning (ML) is a form of artificial intelligence which is placed to transform the twenty-first century. Rapid, recent progress in its underlying architecture and algorithms and growth in the size of datasets have led to increasing computer competence across a range of fields. These include driving a vehicle, language translation, chatbots and beyond human performance at complex board games such as Go. Here, we review the fundamentals and algorithms behind machine learning and highlight specific approaches to learning and optimisation. We then summarise the applications of ML to medicine. In particular, we showcase recent diagnostic performances, and caveats, in the fields of dermatology, radiology, pathology and general microscopy.